Coprime mappings and lonely runners

For x real, let { } $ \lbrace \rbrace$ be the fractional part of (that is, = − ⌊ ⌋ $\lbrace x\rbrace - \lfloor \rfloor$ ). The lonely runner conjecture can stated as follows: for any n positive integers v 1 < 2 ⋯ v_1 v_2 \dots v_n$ , there exists a real number t such that / ( + ) ⩽ i 1/(n+1) \leqslant v_i t\rbrace n/(n+1)$ 1, n$ . In this paper, we prove if ε > 0 \epsilon >0$ and is sufficiently large (relative to ε), then collection v_n (2-\epsilon )n$ This an approximate version natural next step study suggested by Tao. A key ingredient in our proof result on coprime mappings. Let B sets integers. bijection f : → f:A \rightarrow B$ mapping $f(a)$ are every ∈ \in A$ We show ⊂ [ ] $A,B \subset [n]$ intervals length 2m where m e Ω log e^{ \Omega ({(\log \log n)}^2)}$ from B.